On p-primary torsion of the Brauer group in characteristic p

Let k be a finitely generated field. Relation between the Tate conjecture for divisors and finiteness properties of the Brauer groups of varieties over k is well known, at least for torsion coprime to char(k). Much less is known about p-primary torsion in characteristic p. In a recent paper, D'Addezio clarified the situation for abelian varieties over fields of positive characteristic. Using similar ideas, I will show that for varieties X and Y satisfying some mild conditions, the cokernel of the map from Br(X) \oplus Br(Y) to Br(X\times Y) is a direct sum of a finite group and a p-group of finite exponent (which can be infinite). This implies, for example, that the transcendental Brauer group of surfaces dominated by products of curves has finite exponent.